Anti-inverse transformation semigroups

The concept of a anti-regular semigroup was introduced by Jack Sharp in 1977. The name was changed into anti-inverse by Stojan Bogdanovic one year later. We give a survey about the basic properties of anti-inverse semigroups. In this presentation, we will consider anti-inverse transformation semigroups. First, we characterize anti-inverse elemnets in the full transformation semigroup on a finite chain. Then we determine all anti-inverse transformation semigroups within a J-classes. We will specify this results for semigroups of order- (orientation-) preserving (or reversing) transformations.